I'll take a stab at this with the disclaimer that I know absolutely nothing about the laws in Cyprus. I don't know if there is anyone from Nicosia here as well, so it may come down to looking up customs laws online and go from there. What I'll give you is from a US perspective.
I'm working on a manuscript that has my main characters flying into the Larnaca airport in Cyprus. They are general aviation, help with the arrival details would be awesome. (Yes it's fiction but I want to get most of it right.)
1. Would they clear customs planeside?
More than likely not. Here in the US, any airport can be an international airport or port of entry, pending on the availability of customs being able to make it to that airport and perform their checks. Case in point: KSBD (San Bernardino Int'l Airport) here in the US is listed as an international airport, but doesn't have any scheduled international flights coming to it, let alone customs stationed at that airport. If a flight goes there and needs customs to check it, customs makes the drive over from KONT (Ontario Int'l, 10 miles away). The pilots and passengers will not be able to disembark until Customs finishes their checks.
2. Would the FBO (assuming prior arrangements were made) have rental vehicles waiting for them?
I don't see why not. The operations at the FBO will not be affected by Customs. If the customs check fails, then obviously the plane has to depart, and the vehicles will not be used.
3. Don't worry about cost, this is a scenario where money isn't an issue. It's part of a thriller and my operatives are posing as a group of location scouts for a film.
4. I know about live atc because I use it as "background" noise while I write.
5. I'm not a pilot and have no aviation experience save flying on planes all over the globe. Never Cyprus though.
Again, it all just matters on the local laws regarding customs in Cyprus, otherwise, it should be fairly similar across the board.
BL.